On the theoretical front, the problem can be postulated as a magneto-hydrodynamic free-surface flow coupled with heat transfer and phase transitions. Designing self-consistent models or simulations of such systems is very challenging and computationally expensive, especially in cases where the flow is governed by multiple spatial and temporal scales. As a result, compromises must be made depending on which results are sought. For example, if the focus is set on the macroscopic flow features and large-scale melt displacements, simplifications such as the shallow water approximation [1]} have been shown to reproduce the experimental data accurately and at reasonable computational costs [2]}, [3]}, [4]}, [5]}, [6]}. However, the shallow water equations are not suited to study melt ejection, flow instabilities or strong deformations of the free surface. These processes require the solution of the full set of Navier-Stokes equations in a multi-phase framework.
